Civil Engineering Reference
In-Depth Information
provides an increase in strength of the steel it is on the safe side to do so. Finally we
shall assume that both Young's modulus,
E
, and the yield stress,
σ
Y
, have the same
values in tension and compression, and that plane sections remain plane after bending.
The last assumption may be shown experimentally to be very nearly true.
PLASTIC BENDING OF BEAMS HAVING A SINGLY
SYMMETRICAL CROSS SECTION
This is the most general case we shall discuss since the plastic bending of beams of
arbitrary section is complex and is still being researched.
Consider the length of beam shown in Fig. 18.2(a) subjected to a positive bending
moment,
M
, and possessing the singly symmetrical cross section shown in Fig. 18.2(b).
If
M
is sufficiently small the length of beam will bend elastically, producing at any
section mm, the linear direct stress distribution of Fig. 18.2(c) where the stress,
σ
,at
a distance
y
from the neutral axis of the beam is given by Eq. (9.9). In this situation
the
elastic neutral axis
of the beam section passes through the centroid of area of the
section (Eq. (9.5)).
Suppose now that
M
is increased. A stage will be reached where the maximum direct
stress in the section, i.e. at the point furthest from the elastic neutral axis, is equal
to the yield stress,
σ
Y
(Fig. 18.3(b)). The corresponding value of
M
is called the
yield
moment, M
Y
, and is given by Eq. (9.9); thus
σ
Y
I
y
1
M
Y
=
(18.1)
y
y
m
Section mm
M
M
Elastic
neutral
axis
F
IGURE
18.2
Direct stress due
to bending in a
singly symmetrical
section beam
G
G
x
z
m
(a)
(b)
(c)
y
s
Y
s
Y
s
Y
s
Y
y
1
Elastic
neutral
axis
G
z
y
2
Plastic
neutral
axis
F
IGURE
18.3
Yielding of a beam
section due to
bending
s
Y
s
Y
s
Y
(a)
(b)
(c)
(d)
(e)
